Abstract. We prove in this paper the Ax-Lindemann-Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular we reprove a result of Silverberg [57] in a different approach. Then combining these results we prove the André-Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of A n 6 and under GRH for all mixed Shimura varieties of abelian type.
Let
$\mathcal {A} \rightarrow S$
be an abelian scheme over an irreducible variety over
$\mathbb {C}$
of relative dimension
$g$
. For any simply-connected subset
$\Delta$
of
$S^{\mathrm {an}}$
one can define the Betti map from
$\mathcal {A}_{\Delta }$
to
$\mathbb {T}^{2g}$
, the real torus of dimension
$2g$
, by identifying each closed fiber of
$\mathcal {A}_{\Delta } \rightarrow \Delta$
with
$\mathbb {T}^{2g}$
via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety
$X$
of
$\mathcal {A}$
is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char
$0$
and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if
$X$
satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.
In this paper we prove the mixed Ax–Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular, we present an application for studying the generic rank of the Betti map.
Consider a smooth, geometrically irreducible, projective curve of genus g ≥ 2 defined over a number field of degree d ≥ 1. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of g, d, and the Mordell-Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounds, in g and d, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel-Jacobi map. Both estimates generalize our previous work for 1-parameter families. Our proof uses Vojta's approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second-and third-named authors. Contents 1. Introduction 1 2. Betti map and Betti form 7 3. Setup and notation for the height inequality 13 4. Intersection theory and height inequality on the total space 16 5. Proof of the height inequality Theorem 1.6 22 6. Preparation for counting points 23 7. Néron-Tate distance between points on curves 29 8. Proof of Theorems 1.1, 1.2, and 1.4 31 Appendix A. The Silverman-Tate Theorem revisited 36 Appendix B. Full version of Theorem 1.6 40 References 47
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